Groups and topology

PI1: Fernando Muro
PI2: Ramón J. Flores

Abstract: Since their inception as fields of science at the beginning of the XX century, topology and group theory have followed parallel path more often than not. Interaction between them are prevasive. For instance, unstable homotopy theory grew interwoven with finite and compact Lie group theory. The modern development of low-dimensional topology is inseparable from geometric group theory. Higher order sructures such as operads and their algebras derive from algebraic models for loop spaces, which is a topological generalization of the algebraic notion of group. They are currently being applied in other areas such as algebraic geometry and number theory. In this project, our goal is to study a variety of connections between groups and topological spaces and their applications. More precisely, we will study Arting groups via their actions on certain spaces, K-theory of group C-star algebras in connection with the Baum-Connes conjecture, homological knot invariants, models for triangulated categories, and localizations in group theory and topology.